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  • 1
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-1416
    Keywords: Key words: Optimal harvesting ; Stochastic logistic model ; Threshold population density ; Value of the opportunity to harvest ; Itô’s theorem. ; Subject Classification: AMS 60J70 ; 60J60 ; 92D25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract.  We obtain the optimal harvesting plan to maximize the expected discounted number of individuals harvested over an infinite future horizon, under the most common (Verhulst-Pearl) logistic model for a stochastically fluctuating population. We also solve the problem for the standard variants of the model where there are constraints on the admissible harvesting rates. We use stochastic calculus to derive the optimal population threshold at which individuals are harvested as well as the overall value of the population in the sense of the model. We show that except under extreme conditions, the population is never depleted in finite time, but remains in a stationary distribution which we find explicitly. Needless to say, our results prove that any strategy which totally depletes the population is sub-optimal. These results are much more precise than those previously obtained for this problem.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Probability theory and related fields 23 (1972), S. 163-170 
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The equation (*) $$F(qx) = \int\limits_0^x {\left( {1 - F\left( u \right)} \right)du,} {\rm{ }}x \mathbin{\lower.3ex\hbox{$\buildrel〉\over{\smash{\scriptstyle=}\vphantom{_x}}$}} {\rm{0}}$$ where F is a distribution function (d.f.), arises when the limiting d.f. of the residual-lifetime in a renewal process is a scaled version of the general-lifetime d.f. F. The equation (**) $$G(qx) = \int\limits_0^x {uG\left( {du} \right),} {\rm{ }}x \mathbin{\lower.3ex\hbox{$\buildrel〉\over{\smash{\scriptstyle=}\vphantom{_x}}$}} {\rm{0}}$$ on the other hand arises when the limiting d.f. of the total-lifetime in a renewal process is a scaled version of the general-lifetime d.f. G. For 0〈q〈1 the class, F q , of all d.f.'s satisfying (*) has been recently characterized and shown to include infinitely many d.f.'s. By explicitly exhibiting all the extreme points of F q , we recharacterize F q as the convex hull of its extreme points and use this characterization to show that for q close to one the d.f. solution to (*) is “nearly unique.” For example, if q〉0.8 then all the infinitely many d.f.'s in F q agree to more than 15 decimal places. The class, G q , of all d.f. solutions to (**) is studied here, apparently for the first time, and shown to be in a one-to-one correspondence with F q ; symbolically, 1−F q (x) is the Laplace transform of G q (qx). For 0〈q〈1, we characterize Gq as the convex hull of its extreme points and obtain results analogous to those for F q . For q〉1 we give a simple argument to show that neither (**) nor (*) has a d.f. solution. We present a complete, self-contained, unified treatment of the two dual families, G q and F q , and discuss previously known results. A further application of the theory to graphical comparisons of two samples (Q-Q plots) is described.
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  • 6
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Random covering intervals are placed on the real line in a Poisson manner. Lebesgue measure governs their (random) locations and an arbitrary measure μ governs their (random) lengths. The uncovered set is a regenerative set in the sense of Hoffmann-Jørgensen's generalization of regenerative phenomena introduced by Kingman. Thus, as has previously been obtained by Mandelbrot, it is the closure of the image of a subordinator —one that is identified explicitly. Well-known facts about subordinators give Shepp's necessary and sufficient condition on μ for complete coverage and, when the coverage is not complete, a formula for the Hausdorff dimension of the uncovered set. The method does not seem to be applicable when the covering is not done in a Poisson manner or if the line is replaced by the plane or higher dimensional space.
    Type of Medium: Electronic Resource
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  • 7
    ISSN: 1572-9613
    Keywords: Self-avoiding walk ; self-avoiding polygon ; Monte Carlo ; ergodicity ; pivot algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We propose a new class of dynamic Monte Carlo algorithms for generating self-avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension, and prove that these algorithms are ergodic in all cases. We also prove the ergodicity of a variant of the pivot algorithm.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Journal of statistical physics 97 (1999), S. 209-231 
    ISSN: 1572-9613
    Keywords: parking ; probability model ; Laplace transform ; vector differential equation ; explicit solution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The car parking problem is a one-dimensional model of random packing. Cars arrive to park on a block of length x, sequentially. Each car has, independently, spin up or spin down, w.p. 0 〈 p ≤ 1, for spin up and q = 1 − p for spin down, respectively. Each car tries to park at a uniformly distributed random point t ∈ [0, x]. If t is within distance 1 of the location of a previously parked car of the same spin, or within distance a of the location of a previously parked car of the opposite spin, then the new car leaves without parking and the next car arrives, until saturation. We study the problem analytically as well as numerically. The expected number of up spins c(p, a) per unit length for sufficiently large x is neither monotonic in p for fixed a, nor is it monotone in a for fixed p, in general. An intuitive explanation is given for this nonmonotonicity.
    Type of Medium: Electronic Resource
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  • 9
    ISSN: 1572-9192
    Keywords: information theory ; directed graph ; Sperner theorem ; Shannon capacity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Shannon introduced the concept of zero-error capacity of a discrete memoryless channel. The channel determines an undirected graph on the symbol alphabet, where adjacency means that symbols cannot be confused at the receiver. The zero-error or Shannon capacity is an invariant of this graph. Gargano, Körner, and Vaccaro have recently extended the concept of Shannon capacity to directed graphs. Their generalization of Shannon capacity is called Sperner capacity. We resolve a problem posed by these authors by giving the first example (the two orientations of the triangle) of a graph where the Sperner capacity depends on the orientations of the edges. Sperner capacity seems to be achieved by nonlinear codes, whereas Shannon capacity seems to be attainable by linear codes. In particular, linear codes do not achieve Sperner capacity for the cyclic triangle. We use Fourier analysis or linear programming to obtain the best upper bounds for linear codes. The bounds for unrestricted codes are obtained from rank arguments, eigenvalue interlacing inequalities and polynomial algebra. The statement of the cyclic q-gon problem is very simple: what is the maximum size N q(n) of a subset S n of {0, 1, $$\ldots$$ , q−1} n with the property that for every pair of distinct vectors x = (x i), y = (y i) $$\in $$ S n, we have x j −y j ≡ 1(mod q) for some j? For q = 3 (the cyclic triangle), we show N 3(n)≃2 n . If however S n is a subgroup, then we give a simple proof that $$\left| {S_n } \right| \leqslant \sqrt 3 ^n $$ .
    Type of Medium: Electronic Resource
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